L(s) = 1 | − 1.41·2-s + 2.00·4-s − 2.23i·5-s + 11.8i·7-s − 2.82·8-s + 3.16i·10-s + 20.7i·11-s + 7.75·13-s − 16.7i·14-s + 4.00·16-s − 22.9i·17-s + 21.0i·19-s − 4.47i·20-s − 29.3i·22-s + (21.9 − 6.71i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.447i·5-s + 1.68i·7-s − 0.353·8-s + 0.316i·10-s + 1.88i·11-s + 0.596·13-s − 1.19i·14-s + 0.250·16-s − 1.34i·17-s + 1.10i·19-s − 0.223i·20-s − 1.33i·22-s + (0.956 − 0.292i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.440890977\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.440890977\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-21.9 + 6.71i)T \) |
good | 7 | \( 1 - 11.8iT - 49T^{2} \) |
| 11 | \( 1 - 20.7iT - 121T^{2} \) |
| 13 | \( 1 - 7.75T + 169T^{2} \) |
| 17 | \( 1 + 22.9iT - 289T^{2} \) |
| 19 | \( 1 - 21.0iT - 361T^{2} \) |
| 29 | \( 1 - 38.2T + 841T^{2} \) |
| 31 | \( 1 - 33.3T + 961T^{2} \) |
| 37 | \( 1 - 49.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 0.679T + 1.68e3T^{2} \) |
| 43 | \( 1 + 7.99iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 85.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 66.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 110.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 16.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 117. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 28.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 31.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 105. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 43.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 49.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 69.6iT - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.087255664281258043146470224237, −8.580361223044125377344657659887, −7.78078871282195162531569192493, −6.88894866879087311598362436932, −6.11902152692400116365692227141, −5.15493253214656036664491061722, −4.53623599656229171896867985365, −2.95393485840776362747179597334, −2.23079882969989444874537783202, −1.18664174978133481462560904008,
0.55970021844600723507728475626, 1.19203543565429720840188038744, 2.85663145586902031271908072287, 3.57555184280955158485295664558, 4.46084441816531580566177977000, 5.88257012552311644534874626808, 6.46071682313037912759112574167, 7.22114802544061482916361694364, 8.005573167848041308456455042633, 8.634094123530411314564893797857